Stress analysis under centrifugal load of the multiple corrugated diaphragm coupling based on rotating shell thin film model

Centrifugal force is one of the factors that cannot be ignored in high-speed shaft systems. The multiple corrugated diaphragm (MCD) Coupling is suitable for high-power and high-speed situations; hence, it is crucial to investigate the stress and deformation of the wave disc under centrifugal load. This paper first uses the rotating shell thin film model to derive the circumferential stress and deformation displacement of the MCD under centrifugal load. Then, the finite element method is used to verify the results obtained from the analytical solution. The results show the feasibility of using the rotating shell thin film model for centrifugal load analysis of the MCD, providing a new approach for the theoretical analysis of the MCD.


Parameters of the MCD
The shape of the MCD is shown in Fig. 1.The specific geometric and working condition parameters of the MCD in this paper are shown in Table 1.The subsequent analysis process neglects the influence of bolt holes and spline keys.

Circumferential force of the MCD under centrifugal load
Since centrifugal force is an axisymmetric volume load, the MCD under centrifugal action can be regarded as a rotating shell thin film model under the action of axisymmetric volume load.First, the rotating shell thin film model's force balance (differential) equation is derived.Then, the centrifugal loading condition is introduced to   Operating speed (working condition parameter) r/min 5200 find the meridional and latitudinal tension of the rotating shell.Since the structure of the MCD is quite special, its axial section is a complete period of a sine curve.Multiple singular points will appear due to the sudden change in the radius of curvature in the sine curve and two extreme points on the curve.The axial section is analyzed and calculated in the Cartesian coordinate system to avoid the occurrence of singular points and the influence of the radius of curvature.
A rotating shell is formed by a plane curve rotating about an axis coplanar with this curve.This curve is called a meridian, and the plane where the curve is located is called the meridian plane.The horizontal circumference formed by any point on the curve rotating about the axis is called a parallel circle.Figure 2 shows a differential element cut from the rotating shell's middle surface using two adjacent meridians and two adjacent parallel circles.
Because both the meridians and parallel circles are curvature lines of the rotating surface, these two families of curves are taken as coordinate lines.The position of the meridian is determined by the angle θ measured from a certain reference meridian plane.In contrast, the position of the parallel circle is established in the Cartesian coordinate system with the rotation axis as the z-axis and the curve extension direction as the x-axis.Here, r is the curvature radius of the parallel circle.
The coordinate axes are defined as follows.The z-axis is perpendicular to the parallel circle, the x-axis is parallel to the circle, and the y-axis is perpendicular to the XOZ plane.The direction of the coordinate axes is shown in Fig. 2. The external force acting on each unit area of the shell surface has components p x , p y , and p z along the positive direction of the coordinate axes.
Next, the balance of the rotating shell element will be investigated (Fig. 2).Three balance equations will be provided based on the balance conditions of the forces in the direction of the three coordinate axes.These three balance equations determine the unknown internal forces N x , N θ .
First, the tangential force on the infinitesimal body is analyzed: Since the centrifugal load is axisymmetric, the infinitesimal body is not subjected to shear force; the diaphragm undergoes axisymmetric deformation under the action of the load.Therefore, the circumferential force N θ does not change with θ , i.e.: The analysis of the circumferential force on the infinitesimal is shown in Fig. 3. Since the latitude line is parallel to the radius of curvature of the circle, r, the arc length of the infinitesimal meridian is: The arc length is simplified to a straight line in the cross-sectional diagram of the diaphragm.Therefore, the length of the straight line is: The element area is: The force exerted along the latitude line in the direction parallel to the circle: (1) ∂N θ ∂θ = 0.
(  The angle between the circumferential force and the vertical direction is dθ/2 .The two forces are projected onto the x-axis pointing to O point: Since dθ is very small: The sum of the two forces simplifies to: The tangential force on the cross-section within the xoz plane in the infinitesimal tangential force analysis diagram is shown in Fig. 4. Since the circular arc is equivalent to a straight line, the tangential force is oriented along the direction of the straight line, where the negative direction is: The force along the positive direction is: A balance equation in the tangential direction can then be established: The third-order small quantity of the equation is neglected to obtain:  www.nature.com/scientificreports/Considering the normal force, the tangential and circumferential forces are projected onto the normal line to obtain: After simplification: Therefore, the radial balance equation can be written as: The load situation is shown in Fig. 5.The centrifugal load acting on the shell element can be regarded as the product of the distributed load intensity p c and the infinitesimal area S.This load intensity p c can be decom- posed into p z and p y on the curve's normal and tangential directions.Without considering the thickness of the diaphragm, the centrifugal force is ρSω 2 x , the centrifugal load per unit area is ρω 2 x , the density of the rotating shell is ρ , and the angular velocity of the diaphragm is ω .Then, the relationship formula for the centrifugal load is: The centrifugal load and the support condition of the shell are axisymmetric.Therefore, the internal force and deformation of the shell must be symmetrical to the rotation axis.Simultaneously, the shear force must be zero; otherwise, the shear force will cause non-axisymmetric deformation.Moreover, other internal stresses do not change with θ ; otherwise, the condition of axisymmetry will be violated.The centrifugal load is substituted into Eqs.( 13) and ( 16) to obtain: The tangential balance equation is: Equation ( 21) is a first-order linear differential equation with constant coefficients; the solution to the differential equation is: Parameter N θ is substituted into Eq.( 20) to obtain: Both ends of the straight line model are processed, the impact of d α on the model is considered, and a balance equation is established in the tangential direction: The increment of straight-line length is decomposed into horizontal and vertical directions.The total change in the straight-line length is divided by the initial length dx cos α of the unit to obtain the strain in the meridian direction of the inner shell: The displacement deformation for a unit of parallel circles is shown in Fig. 6.The increment of the radius r of the circle is u due to displacements v and u.Since the increment of the circumference of the parallel circle is proportional to the increment of the radius, the circumferential strain is: According to Hooke's law, the strains ε x and ε θ are expressed in terms of stresses N x and N θ as follows: Hooke's law is substituted into the circumferential strain ε θ : Diaphragm thickness is introduced since it is related to displacement.The centrifugal load can then be expressed as:

Substituting the expression into stress yields
The displacement is obtained as: (39) du cos α + dv sin α .

Eh
)dx + C is substituted into Eq. ( 50 and integrated: The outer ring of the corrugated diaphragm is fixed.Therefore, according to the boundary conditions, when x = 162 (Outermost ring of MCD profile), the vertical displacement v = 0. Hence, the constant C can be calculated as: With the vertical direction as positive, the vertical displacement equation v is:

Finite element calculation of the MCD under centrifugal load
Since MCD is an axisymmetric structural unit, the axisymmetric finite element analysis method is applicable.When creating the finite element model, only a single section of the MCD needs to be established, greatly reducing the resource usage during analysis and improving the efficiency of analysis; the impact on the accuracy of the analysis is insignificant.The finite element model of MCD is established by the axisymmetric analysis method; only a single section of MCD is established.The volume element type is Solid185.Considering the model structure and calculation accuracy requirements, many sections are established on the model before grid division.Hence, hexahedral elements can be used for grid division.Elastic modulus, Poisson's ratio, and material density were set, the grid was divided into Smart Size (10), the analysis model was initially divided, and the local cell grid was refined; more than 80,000 cells were obtained 4 .
The finite element model of the MCD is established in ANSYS software, as shown in Fig. 7.All degrees of freedom are constrained at the inner hole spline, and the angular velocity around the Y-axis is applied.The structural and load parameters of the MCD are substituted to obtain a solution.Then, the circumferential force and deformation displacement of the waveform diaphragm under centrifugal load can be obtained.www.nature.com/scientificreports/ The results obtained by the finite element method have certain fluctuations in both stress and displacement.According to the further analysis of the MCD profile curve, when the profile radius is 118.5 mm, the MCD profile is at the peak position, where the stress and displacement fluctuate upward relative to the analytical solution.When the profile radius is 147.5 mm, the profile is at the trough position, where the stress and displacement fluctuate downward relative to the analytical solution.The difference between the analytical solution and the finite element results is closely related to the wavy film disk profile.The model is simplified, and the tangential force under the centrifugal force is neglected when the analytical solution is calculated.Under the centrifugal force load, the MCD produces additional deformation under the action of internal stress due to the special profile structure of the MCD.Consequently, tangential stress appears, leading to a fluctuation of the stress and displacement of the MCD relative to the theoretical value.However, because the mechanism of internal stress generation is relatively complicated, the analytical solution cannot be solved if the influence of tangential force generated by internal stress is considered.Therefore, optimizing the model cannot eliminate the fluctuation difference between the two results in this paper.2][3][4][5] , the stress generated by the MCD under centrifugal load is relatively small compared with that generated by other working conditions.In contrast, the maximum stress is only 6 MPa.
In summary, although there are some differences between the analytical and the finite element methods, introducing the rotating shell film model into the theoretical analysis of the MCD is still valid, providing a new way for the theoretical analysis of the MCD.

Conclusions
This paper used the rotating shell thin film model to derive the corresponding relationship between the circumferential force and deformation displacement of the MCD and the centrifugal load.Then, the finite element method was used to analyze the circumferential force and deformation displacement of the MCD under the centrifugal load.The results obtained by the two methods are within the error range, proving the feasibility of introducing the rotating shell thin film model into the theoretical analysis method of the MCD.Consequently, a new method for the theoretical analysis of the MCD coupling was provided.
Although the results of the two methods in this paper are consistent within the error range, the error is still too large.In the future, a coupling stress test bench will be built, the stress test data will be combined, the analytical solution will be optimized to solve the equation and the finite element analysis model, and the results obtained by the two analysis methods will be reduced without affecting the force analysis.Thus, the development and application of MCD coupling will be promoted.

2 9
The radius of the inner circle r0/mm 89 10 The radius of the outer circle r1/mm 182 11 The number of diaphragms in an assembly m/piece 15 12

Figure 3 .
Figure 3. Circumferential force analysis diagram of a microelement.

Figure 7 .
Figure 7. Finite element model of the MCD.

Figure 10 .
Figure 10.The vertical displacement curve obtained by the analytical solution and the finite element method.

Table 1 .
Geometric parameters of the MCD.